Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://www.sagemath.org/pdf/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].

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Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].

Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for.

Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for.

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'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The final exam date and location will be announced by the University and posted here when available.

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'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on November 9, in class. The final exam will be on 12/21/2017, 12:25-2:25PM in Social Sciences building, room 6240.

Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.

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Homework is due at the beginning of class on the specified Thursday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.

Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such N.

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a) Can you formulate a conjecture about the relationship between f(N) and N/log N?

a) Can you formulate a conjecture about the relationship between f(N) and N/log N?

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Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.

Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.

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* '''Sep 17''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19

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* '''Sep 28''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19

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* '''Sep 24''': 2.15, 2.16 (note that I presented part a) of this in class), 2.20, 2.23, 2.26.

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* '''Oct 5''': 2.15, 2.18, 2.23, 2.26, 2.30.

Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.

Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.

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* '''Oct 12''': Book problems: 3.4, 3.5, 3.6

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* '''Oct 1''':

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Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.

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Book problems: 2.31 (I will give a hint for this problem later in the week.) 3.4,3.5,3.6

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Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.

Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.

Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.

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Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?

Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?

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Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)

Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)

Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.

Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.

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* '''Oct 8''':

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* '''Oct 19''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.

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Problem A. Give a prime factorization of the Gaussian integer 7+9i.

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Problem B. We showed in class that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d).

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Problem B. Give a prime factorization of the Gaussian integer 7+9i.

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B.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).

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Problem C. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d).

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C.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).

(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?

(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?

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B.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.

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C.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.

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Problem C. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)

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Problem D. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)

Now define phi(d) to be the number of elements s of S such that s and d are coprime.

Now define phi(d) to be the number of elements s of S such that s and d are coprime.

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C.1. Compute phi(1+2i) and phi(3).

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D.1. Compute phi(1+2i) and phi(3).

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C.2. Prove that the value of phi(d) does not depend on the choice of S.

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C.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)

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D.2. Prove that the value of phi(d) does not depend on the choice of S.

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*'''Oct 15:'''

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D.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)

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4.1

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*'''Nov 2:''' 4.1 from the book.

Problem A. Express 50005 as the sum of two squares.

Problem A. Express 50005 as the sum of two squares.

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Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]

Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]

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D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are primes in Z[sqrt(-d)].

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D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].

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D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into primes. (Hint: imitate the example we used for Z[sqrt(-5)].)

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D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)

''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!

''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!

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Problem E. This is a version of the proof we gave in class that the Diophantine equation y^2 = x^3 - 1 has only the solution x=1,y=0. In this problem we consider the equation y^2 = x^3 - 4.

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*'''November 16'''

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E.1. We can rewrite the equation as (y-2i)(y+2i) = x^3. Show that, if the greatest common divisor of (y-2i) and (y+2i) is not 1, then both y-2i and y+2i are multiples of 1+i.

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Book problems: 5.3,5.4,5.5

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E.2. Prove that (y-2i) and (y+2i) are relatively prime if x is odd.

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E.3. Using this argument, show that the only solutions to y^2 = x^3 - 4 with x odd is (11,5).

Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).

Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).

A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.

A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.

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A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.

A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.

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Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.

Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?

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Problem A. Let p = 58741. Use the method of section 4.5 (discussed in class) to find an r in (Z/pZ) with r^2 = -1.

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Problem B. Use the result of problem A, and the method of section 5.7, to find integers a,b such that a^2 + b^2 = p.

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Problem C. As discussed in class, Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?

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'''November 12'''

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'''December 7'''

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Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.

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Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.

Problem A. When p is a prime congruent to 1 mod 4, prove that ((p-1)/2)! is a square root of -1 in (Z/pZ), along the lines described on the midterm (or by some other means, if you prefer.)

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Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class.

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Problem B. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.

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Problem C. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.

Problem E. An ''ideal'' of Z[i] is a subset I of Z[i] which is closed under addition (if x and y are in I, then x+y is in I) and multiplication by Gaussian integers (if x is in i, then zx is also in I for every Gaussian integer z.) The set of multiples of a Gaussian integer is always an ideal (you don't need to prove this.) List all the ideals of Z[i] containing 2Z[i]. (Because such ideals satisfy conditions 1 and 2 from Monday's lecture, this list will be a subset of the list of 5 subgroups we computed in class.)

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We will not go any further with the notion of ideals in Math 567, but it is worth saying that the language of ideals is absolutely essential for the understanding of contemporary number theory, in a sense "making up for" the failure of unique factorization.

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Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.

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'''November 19'''

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<!-- '''November 19'''

Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2.

Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2.

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OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz? -->

OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz? -->

Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook Elementary Number Theory: Primes, Congruences, and Secrets, which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, it is available as a free legal .pdf. We will be using the (free, public-domain) mathematical software SAGE, developed largely by Stein, as an integral component of our coursework. There is a useful online tutorial. You can download SAGE to your own computer or use it online.

Course Policies: Homework will be due on Thursdays. It can be turned in late only with advance permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.

Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for.

Grading: The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on November 9, in class. The final exam will be on 12/21/2017, 12:25-2:25PM in Social Sciences building, room 6240.

Syllabus:
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)

Homework:
Homework is due at the beginning of class on the specified Thursday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.

Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x.

a) Can you formulate a conjecture about the relationship between f(N) and N/log N?

b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)

c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.)

d) Give as good an upper bound as you can for f(N).

Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.

Sep 28: 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like this to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19

Oct 5: 2.15, 2.18, 2.23, 2.26, 2.30.

Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.

Oct 12: Book problems: 3.4, 3.5, 3.6

Problem A. Prove that there are infinitely many primes p such that 2 is not a primitive root in Z/pZ. We break this up into steps.

Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.

Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?

Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)

Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.

Oct 19: Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.

Problem B. Give a prime factorization of the Gaussian integer 7+9i.

Problem C. Read the notes from here. Note that Z[i] satisfies a reduction theorem: if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d).

C.1. When d = 1+2i, show that, for each n in Z[i], there is a unique pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?

C.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.

Problem D. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)

Now define phi(d) to be the number of elements s of S such that s and d are coprime.

D.1. Compute phi(1+2i) and phi(3).

D.2. Prove that the value of phi(d) does not depend on the choice of S.

D.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)

Nov 2: 4.1 from the book.

Problem A. Express 50005 as the sum of two squares.

In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8 (as shown on the midterm.)

Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is unbounded.

Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?

Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]

D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].

D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)
Remark: The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!

November 16

Book problems: 5.3,5.4,5.5

Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).

A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.

A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.

Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.

Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?

December 7

Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). OPTIONAL: Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.

Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.

Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class.

Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.

December 12 Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.